Mathematics • Year 10 • Unit 1 • Lesson 18

Volume of Prisms and Cylinders — Skill Drill

Build fluency with the universal volume formula from Lesson 18: V = A_base × h. Apply it to rectangular prisms (V = lwh), triangular prisms (V = ½bh × L) and cylinders (V = πr²h), and convert between cm³, m³, mL, L and kL.

Build · I Do / We Do / You Do

1. I do — fully worked example

Read every step. Each one has a short reason on the right so you can see why, not just what.

Problem. A triangular prism has a base triangle with base 8 cm and perpendicular height 5 cm. The prism is 12 cm long. Find its volume in litres.

Step 1 — Spot the rule.

Any prism → V = A_base × h.

Reason: the universal prism formula. For a triangular prism, A_base is the triangle area; h is the prism length.

Step 2 — Calculate the triangle's area.

A_base = ½ × 8 × 5 = 20 cm²

Reason: area of a triangle = ½ × base × perpendicular height. Use the perpendicular 5, not a slant.

Step 3 — Multiply by the prism length.

V = 20 × 12 = 240 cm³

Reason: the cross-section is constant for the full 12 cm length, so the volume is base area × length.

Step 4 — Convert cm³ to litres.

240 cm³ = 240 mL = 0.24 L

Reason: 1 cm³ = 1 mL and 1000 mL = 1 L.

Answer: V = 0.24 L.

Stuck? Revisit lesson § "Volume of Prisms" — Worked Example 1.

2. We do — fill in the missing steps

Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks

Problem. A cylindrical water tank has diameter 1.2 m and height 1.5 m. How many litres of water does it hold when full?

Step 1 — Spot the rule: a cylinder is a prism with a __________________ base, so V = πr²h.

Step 2 — Convert diameter to radius:

r = 1.2 ÷ ______ = ______ m

Step 3 — Substitute into V = πr²h:

V = π × (______)² × 1.5 = π × ______ × 1.5 = ______π m³

Step 4 — Evaluate as a decimal (2 d.p.):

V ≈ ______ m³

Step 5 — Convert m³ to litres (1 m³ = 1000 L):

Capacity ≈ ______ × 1000 ≈ ______ L (nearest litre)

Stuck? Revisit lesson § "Volume of Cylinders" — Worked Example 2. Don't forget to halve the diameter first.

3. You do — independent practice

Show your working in the space under each problem. The first four are foundation (single shape, single unit). The middle two are standard (unit conversion or radius from diameter). The last two are extension (multi-step in syllabus).

Foundation — single-shape volumes

3.1 Find the volume of a rectangular prism with dimensions 6 cm by 5 cm by 4 cm. 1 mark

3.2 Find the volume of a cylinder with radius 3 cm and height 7 cm. Leave your answer in terms of π. 1 mark

3.3 Find the volume of a triangular prism with base triangle (base 10 cm, perpendicular height 6 cm) and prism length 15 cm. 1 mark

3.4 Convert 2500 cm³ to litres. 1 mark

Standard — diameter + conversion

3.5 A rectangular prism has dimensions 15 cm by 10 cm by 8 cm. Find its volume in (a) cm³, and (b) litres. 2 marks

3.6 A cylinder has diameter 8 cm and height 12 cm. Find its volume in cm³ to 1 d.p. 2 marks

Extension — push your thinking

3.7 A swimming pool is 25 m by 10 m with an average depth of 1.8 m. How many kilolitres of water does it hold? (Hint: 1 m³ = 1 kL.) 3 marks

3.8 Two cylinders have the same height of 10 cm. Cylinder A has radius 4 cm; Cylinder B has radius 8 cm. By what factor is Cylinder B's volume larger than Cylinder A's? Show your working. 2 marks

Stuck on 3.8? Doubling the radius squares the multiplier on volume. The volume scales by r², so doubling r gives ×4.

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Answers — Do not peek before attempting

Section 2 — We do (water tank d = 1.2 m, h = 1.5 m)

Step 1: circular base.
Step 2: r = 1.2 ÷ 2 = 0.6 m.
Step 3: V = π × (0.6)² × 1.5 = π × 0.36 × 1.5 = 0.54π m³.
Step 4: V ≈ 1.70 m³ (2 d.p.; more precisely 1.6965).
Step 5: Capacity ≈ 1.70 × 1000 ≈ 1696 L (nearest litre).

3.1 — Rectangular prism (6 × 5 × 4)

V = 6 × 5 × 4 = 120 cm³.

3.2 — Cylinder (r = 3, h = 7)

V = π × 3² × 7 = 63π cm³ (≈ 197.9 cm³).

3.3 — Triangular prism (base 10, ht 6, length 15)

A_base = ½ × 10 × 6 = 30 cm². V = 30 × 15 = 450 cm³.

3.4 — 2500 cm³ to L

2500 cm³ = 2500 mL = 2.5 L (1000 cm³ = 1 L).

3.5 — Rectangular prism 15 × 10 × 8

(a) V = 15 × 10 × 8 = 1200 cm³.
(b) 1200 cm³ = 1200 mL = 1.2 L.

3.6 — Cylinder (d = 8, h = 12)

r = 8 ÷ 2 = 4 cm. V = π × 4² × 12 = 192π ≈ 603.2 cm³ (1 d.p.).

3.7 — Swimming pool 25 m × 10 m × 1.8 m

V = 25 × 10 × 1.8 = 450 m³ = 450 kL.
1 m³ = 1000 L = 1 kL, so m³ and kL are numerically equal.

3.8 — Two cylinders (same h, r = 4 vs r = 8)

V_A = π × 4² × 10 = 160π cm³.
V_B = π × 8² × 10 = 640π cm³.
Ratio = 640π ÷ 160π = 4. Cylinder B is 4 times larger.
Doubling r squares the multiplier on V (since V ∝ r²): 2² = 4.